“On Strategic Advantages of Sales Maximization in a Duopoly”
Charles E. Hegji and Cheng-Zhong Qin*
February 9, 2004
Abstract
This paper applies a two-stage model to study the prevalence of sales-maximizing
behavior of duopolistic firms. In stage one, firms independently and simultaneously
choose as objectives for deciding outputs and/or prices how much weight to place on
profit and how much weight to place on sales. Firms then pursue their chosen objectives
in a duopoly game in stage two. Firms commit to pursuing their objectives through
contracts with input suppliers. The selection of profit versus sales maximization is
analyzed under several common duopoly games in stage two. (JEL L1, L2, C7)
*Hegji: Department of Economics, Auburn University Montgomery, Montgomery, Al
36124 (email: [email protected]). Qin: Department of Economics, University of
California, Santa Barbara, CA 93106-9210 (email: [email protected]). We thank Ted
Bergstrom, Ted Frech, and Christopher Proulx for their helpful comments.
1. Introduction
The classical operating assumption for firm behavior is that firms maximize
profit. However, an alternative assumption, that firms maximize sales, or at times appear
to, has received attention in the literature. One argument as to why firms might
maximize sales is that revenue, particularly revenue in the short run, may be linked to
other firm objectives besides profits. Baumol (1958) and Peston (1959) argue that a high
volume of short-run sales might enhance the ability of a firm to obtain funds in capital
markets, thereby increasing its long-run rate of growth and profits. Amihud and Kimin
(1979) and Vickers (1985) argue that sales may be a more easily achieved and visible
management objective than profits. Along these lines, one would expect to observe sales
maximization as the primary firm objective in situations where there is a high degree of
separation between firm ownership and control.
Fershtman and Judd (1987) formally model the separation of ownership and
control in a duopoly.1 Their model envisions firms as competing through managers in a
two-stage game. In stage one, firm owners compete by designing payment contracts with
managers who will play the subsequent duopoly game in stage two on behalf of the
owners. In their model, managers’ objectives, which are to maximize payments specified
in the payment contracts, become the objectives of the firms in the duopoly game.
Fershtman and Judd examine how the resulting equilibrium price and outputs are affected
by such delegation of control. An important contribution is that the authors demonstrate
that competing through managers makes firms more aggressive, in the sense that owners
have incentives to direct managers away from profit maximization.
2
In this paper, we model a mixed profit and sales oriented objective as a strategic
move 2 that once credibly made enables an owner-managed firm to become more
aggressive in its output and/or pricing decisions than directly pursuing profit
maximization. We study firms’ selections of such mixed profit and sales objectives and
their impacts on equilibrium output and/or price decisions using a two-stage approach. In
our model’s first stage, firms simultaneously and independently choose how much weight
to attach to profit and how much weight to attach to sales. In the second stage, firms
pursue their chosen objectives in a duopoly game.
In Fershtman and Judd’s (1987) model, decisions on output and/or price in a firm
are delegated to a manager, so that the firm can credibly commit to pursue objectives
different from profit maximization in a duopoly game through proper payment contracts
between the firm and the manger. Payments to managers are in the form of a weighted
sum of profit and sales plus a constant, where the weights on profit and sales are not
restricted to be nonnegative. Fershtman and Judd show that owners have incentives to
penalize for profits by having negative weights on profit in payment contracts when costs
are low (see Theorem 1 in Fershtman and Judd, 1987, pp.932). In contrast, firm owners
in our model operate without managers. Consequently, we cannot consider payment
contracts with managers as a commitment device for firms to pursue mixed profit and
sales objectives. Instead, owners may achieve such commitments through factor supply
contracts.
1
The major analysis in Fershtman and Judd (1987) pertains to a duopoly.
Schelling (1960, pp. 160) defines a strategic move as one that influences the other person's choice in a
manner favorable to one's self, by affecting the other person's expectations of how one's self will behave.
2
3
With a mixed profit and sales objective, a firm maximizes profit on the output
market as if its marginal cost is scaled down by the weight on profit. It follows that the
firm can credibly commit to pursue the mixed objective if it can in fact reduce its
marginal cost to the corresponding level. One way to do this is for the firm to transform a
portion of its variable cost to fixed cost. For example, the firm may approach its input
supplier with a two-part payment contract offering to pay a fixed fee and a per unit factor
cost that yields the reduced marginal cost.3 Such a scheme could not yield negative
marginal costs, since that would require the input supplier to pay the firm on a per unit
basis for resources purchased, which from a practical point of view is absurd.4
Consequently, with two-part payment contracts between firms and input suppliers as the
commitment device, it is natural to restrict mixed objectives to only involve nonnegative
weights on profit and sales. We will further elaborate on this commitment device in the
next section.
This shifts our emphasis away from managerial incentives to that of determining
strategic advantages of pursuing mixed profit and sales objectives. A unique feature of
our paper is to demonstrate the advantages of pursuing such mixed objectives in terms of
after play profits and to link these advantages to firm costs. Our results show that when a
firm’s cost is small relative to demand and when competition on the output market is in
Cournot style, sales maximization is dominant for the firm. Given any objective selection
by the rival, a firm is always better off committing to pursue some mixed profit and sales
3
As an example, automobile makers in Detroit often sign contracts with labor unions that require them to
pay wages even if the workers are laid off. The effect of this pre-commitment is that the cost of labor does
not depend on firm output level and should therefore be treated as a fixed cost. In essence the precommitment increases fixed cost and decreases variable cost, but not to a negative value. Such severance
pay agreements are also prevalent in the steel industry and between workers and firms in some nonmanufacturing sectors such as public utilities (see Pita 1996).
4
objective than committing to pursue direct profit maximization. Equilibrium with a
strictly mixed profit and sales objective for both firms exists under certain demand and
cost conditions. Furthermore, when restricting objectives to be either profit maximization
or sales maximization, firms are trapped in a prisoner’s dilemma with objective selections
under high demand and low costs conditions. Other conditions on demand and costs yield
either a chicken game or a situation in which profit maximization dominates sales
maximization.
With Stackelberg competition in the output market, there is a difference between
the behavior of the Stackelberg leader and follower. While conditions exist under which
sales maximization is dominant for the Stackelberg follower, the Stackelberg leader is
always better off directly pursuing profit maximization. The dominance of profit
maximization over other objectives also holds for duopolistic firms engaging in Bertrand
competition in the output market. Finally, our paper also considers a strategic entry
deterrent advantage of sales maximization. The upshot of this analysis is that when postentry competition is in Cournot style, sales maximization also has a market deterrent
advantage for an incumbent firm.
The rest of the paper is organized as follows. Section 2 begins with the set up of our
basic model and presents results under Cournot, Stackelberg, and Bertrand competition in
the output market. Section 3 studies the market deterrent advantage of sales maximization
for an incumbent. Conclusions are outlined in Section 4. All proofs are collected in the
Appendix.
4
The idea to consider factor supply contracts as strategic commitments has been discussed by Krouse
5
2. Sales versus Profit Maximization
In this paper a mixed profit and sales oriented objective is to maximize a weighted
sum of profit and sales. Without loss of generality, we restrict the weights on profit and
sales to sum to one. Thus we may identify a mixed profit and sales objective with the
weight on profit. Accordingly, by objective λi for firm i we means that firm i has
committed to guide its output and/or price decisions in stage two by the following
objective function:
OBJ i  i [ Pqi  Ci (qi )]  (1  i ) Pqi  Pqi  i Ci (qi ) for i 1, 2,
(1)
where P denotes the price, Ci(qi) denotes firm i’s cost function. Observe that (1) implies
that pursuing a mixed profit and sales objective is equivalent to maximizing profit with
cost function discounted by a factor equal to weight λi. How can firm i credibly commit
itself to maximize objective function (1) when its cost function is Ci(qi)?
To find an answer, we assume that firms’ marginal costs are constant and fixed
costs are zero. That is, assume C1(q1) = C1q1, C2(q2) = C2q2. Suppose that
objectives 1 and  2 have been selected. Consider a contractual arrangement between firm
i and its input supplier, under which firm i pays both an irrevocable fixed fee F in
advance and a per unit cost λiCi. We assume that there is only one composite factor of
production and units are scaled so that its marginal physical product is unity. Suppose
that the factor supplier has the same common knowledge as firm i and has the same
computational skills. Then given objective selections 1 and  2 , the supplier can
anticipate Nash equilibrium of the subsequent duopoly game, and will be indifferent
between charging firm i according to the two-part contract and the single price Ci so long
(1990, pp. 378-381) who attributes the discussion to Maksimovic (1984).
6
as F + λiCi qi*  C i qi* , where qi* is firm i’s output in Nash equilibrium of the
subsequent duopoly game. The arrangement can thus be thought of as a way by which
firm i can credibly lower its variable cost by transforming some of its variable cost into
fixed costs, holding total costs in equilibrium constant (see Krouse, 1990, pp. 378-381).
We note here that with the above contractual arrangement we can still use firm i’s
original marginal cost Ci to calculate firm i’s profit in Nash equilibrium of the subsequent
duopoly game.
We solve the two-stage game via backward induction: First find Nash equilibrium
for the duopoly game in stage two for given objective selections. Then compute firms’
reduced-form profits as functions of objective selections. Finally, solve for firms’ optimal
objective selections by finding Nash equilibrium for the reduced-form game between
firms, in which payoffs are reduced-form profits and strategies are objective selections.
The resulting solution is a subgame-perfect equilibrium. Our major analysis focuses on
the reduced-form game. In particular, by a dominant objective selection for a firm we
refer to the dominant strategy for the firm in the reduced-form game.
2.1. Stage Two Competition Is Cournot
Consider a situation with a homogenous good produced by two firms, called firm
1 and firm 2. The inverse demand function will be given by P = A – BQ with Q = q1 +
q2, where P is price and qi is quantity produced by firm i. Parameters A and B are both
positive. Demand is assumed to be linear for reasons of tractability. In addition, firms
have positive constant marginal costs, C1 for firm 1 and C2 for firm 2, and zero fixed
costs. All parameters are known to both firms.
7
Suppose that firms play a Cournot duopoly game in stage two given objective
selections in stage one. That is, given λ1 and λ2 , firms play a quantity setting game in
stage two, in which firm i’s payoff function is as in (1). With interior solution, simple
calculation shows that firm i’s output reaction function in stage two is
qi (q j ) 
1  A  i C i

 q j , i, j 1,2, i  [0,1].

2 B

(2)
Assume 2C1 < A and 2C2 < A. These conditions guarantees that the Cournot (Nash)
equilibrium outputs for the two firms and market price will be
qi* 
A   j C j  2i Ci
3B
, i  j, P * 
A  1C1  2 C 2
.
3
(3)
Note that the equilibrium price increases with the values of λ1 and λ2. Quantities
and price in (3) determine firms’ profits associated with objective selections =(1, 2).
This way, we have derived the reduced-form game between firms, in which strategies are
objective selections and payoffs are profits associated with such objective selections. We
note here that firms’ objective selections in Nash equilibrium of this reduced-form game
are their objective selections in subgame-perfect equilibrium of the two-stage game.
THEOREM 1: Suppose stage two competition is Cournot and suppose A > 2C1 and A >
2C2. Then,
(i)
Sales maximization is dominant for firm i if and only if 6Ci  A.
(ii)
For any 0   j  1, there exists 0  i  1 such that firm i makes more
reduced-form profit with objective i than with profit maximization.
8
(iii)
There exits a Nash equilibrium for the reduced-form game with a strictly
mixed profit and sales objective for each firm if and only if 8C i  A  2C j
and 3C i  A  2C j .
Theorem 1 summarizes strategic advantages of sales maximization over profit
maximization for a firm under various demand and cost configurations, given that
competition in the second stage is in Cournot style. In general, sales maximization
requires firms to aggressively bit down price in an attempt to increase market share.
Point (i) suggests that when 6Ci  A, this aggressive strategy will be dominant for firm i.
Point (ii) implies that firms always attempt to implement some amount of sales
maximization. Point (iii) furthermore specifies conditions under which Nash equilibrium
for the reduced-form game (subgame-perfect equilibrium for the two stage game) exists,
in which firms place positive weights on both sales and profit maximization. These
conditions also imply that owners in Fershtman and Judd’s (1987) model choose positive
weights on both profit and sales in their payment contracts with managers in subgameperfect equilibrium.
Results in Theorem 1 are in the spirit of the findings of Rhodes and Stegeman
(2001), who use an evolutionary game model to demonstrate that under high demand and
low cost firm behavior is distorted toward sales maximization. Rhodes and Stegeman see
this as the outcome of a process in which firms imitate the strategies of other firms
yielding the largest profits. In contrast, our analysis focuses directly on the strategic
aspect of sales maximization. A central idea conveyed in Theorem 1 is that in strategic
situations players may benefit from pursuing this type of distorted objective. This idea is
complementary to recent work of Heifetz, Shannon, and Spiegel (2002).
9
We analyze next a particular simplified case in which firms only consider as
objectives either to maximize profit or to maximize sales but not any mixture of the two.
2.1 A. The Possibility of a Prisoner’s Dilemma or Chicken with Pure Profit and Sales
Maximization
When restricting the objective selections to be either profit maximization or sales
maximization, conditions on demand and costs can be found for the reduced-form game
to take on the properties of various specific games. We summarize these results in the
following theorem.
THEOREM 2: Suppose stage two competition is Cournot and A > 2C1 and A > 2C2.
Suppose further the weight on profit is restricted to be either 0 or 1. Then,
(i) Sales maximization is dominant for firm i if and only if 4Ci < A.
(ii) Firms are trapped in Prisoner’s Dilemma if and only if 4Ci < A and A(2Cj – Ci) +
(Cj – Ci)2 > 0 for i ≠j.
(iii) Firms face a Chicken Game if and only if A < 4Ci < A + Cj for i ≠ j.
(iv) Profit maximization is dominant for firm i if and only if 4Ci >A + Cj for i ≠ j.
Point (i) of Theorem 2 shows that similar to the continuous case, in the discrete
case of maximizing profit versus maximizing sales, maximizing sales is dominant under
conditions of high demand and low cost. Note, however, that since the rival is restricted
in its objective, the condition on demand relative to cost under which maximizing sales is
dominant is less restrictive than the continuous case. Point (iv) of Theorem 2 shows that
under an alternative configuration of demand and costs, firms will gravitate toward profit
maximization. The conditions 4C1 > A + C2 and 4C2 > A + C1 may loosely be
interpreted to represent a situation of low demand and high costs.
10
The sense in which sales maximization is dominant under Cournot competition is
further clarified by points (ii) and (iii). If the conditions in point (ii) are satisfied, each
firm’s profits would be larger if firms commit in stage one to maximize profits. However,
such a commitment by any firm is not credible; profit maximization is dominated for
each firm. This is a classic Prisoner’s Dilemma situation, where each of the two firms
could capture substantial gains through cooperation (to pursue profit maximization) but is
tempted by even greater gains to act selfishly (to pursue sales maximization), while the
other firm acts cooperatively. In passing, note that when firms have equal marginal costs,
that is C1 = C2 = C, the latter condition in point (ii) reduces to AC + C2 > 0. This
condition always holds as long as C > 0. Therefore, if costs for the two firms are equal,
and costs and demand are such that sales maximization is dominant for both firms, the
firms will necessarily be trapped in a Prisoner’s Dilemma.
Another point to consider is that although the firms collectively earn less profit by
both maximizing sales than by both maximizing profits, consumers are better off.
Indeed, total consumer surplus when firms maximize sales is 4A2/18B, as compared to
(2A – C1 –C2)2/18B in the profit maximizing equilibrium. Furthermore, with linear cost
and demand functions, the fact that mutual sales maximization results in a lower price
than mutual profit maximization implies that total surplus (consumer surplus plus
producer surplus) is higher with the former than with the latter. This means that as a
policy implication, when competition in an industry is in Cournot style with low cost and
high demand, it is more socially efficient not to regulate the industry, because regulation
increases firm’s costs which may make profit maximizing dominant.
11
Under conditions in point (iii) there are two possible Nash equilibrium in pure
strategy, * = (1, 0) and * = (0, 1) in the reduced-form game. The situation is a game of
Chicken where both participants place a high cost (i.e., low profit) to being designated
“Chicken.” In this case, the Nash equilibrium profit with sales maximization is higher
than that with profit maximization. As is well known, in addition to the two pure-strategy
Nash equilibria in a chicken game, there is also Nash equilibrium in mixed strategies. The
mixed strategy Nash equilibrium under condition (iii) in Theorem 2 has some interesting
interpretations for our model. We summarize the mixed strategy Nash equilibrium in
Theorem 3 below.
THEOREM 3: Suppose stage two competition is Cournot and suppose A < 4C1 < A + C2
and A < 4C2 < A + C1. Suppose further the weight on profit is restricted to either 0 or 1.
Then, there is a Nash equilibrium in mixed strategies for the reduced-form game with
firm 1 choosing profit maximization with probability 1* and firm 2 choosing profit
maximization with probability 2*, where 5
1* 
4C2  A
4C  A
and  2* 1
.
C1
C2
Note that the restrictions on A, C1, and C2 guarantee 3Ci < A and 0 < i* < 1 for i
=1, 2. Also note that as a firm’s rival’s marginal cost increases, the firm pursues profit
maximization with greater frequency. When firms have the same marginal costs, C1=C2
=C, the two firms choose profit maximization with equal frequency. However, this
frequency is not necessarily 50% of the time, but varies with cost and demand. With
5
These probabilities in general differ from the equilibrium weights placed on profits and sales because of
the criteria by which they are determined.
12
equal costs for the two firms, sales and profit maximization are chosen with equal
frequency only when A = 7C/2.
We also note that under the conditions on demand and costs in Theorem 3,
∂  * i/∂Ci = (A – 4Cj) /Ci2 < 0 since A < 4Cj and ∂  * i/∂Cj = 4 / Ci  0 . This means that as
firm i’s marginal cost decreases or as firm j’s marginal cost increases, the probability
attached to profit maximization increases for firm i. This is apparently not intuitive
because firm i becomes more aggressive in the sense that it tends to produce more when
its marginal cost decreases. The reason has to do with the notion of equilibrium in mixed
strategies (e.g. Kreps, 1990, pp. 407-410). The primary objective of a mixed strategy is
to neutralize the opponent’s strategy by equalizing the rival’s expected payoff under all of
the rival’s strategy options. Assume i= 1 and j = 2. Examination of firm 2’s profit reveals
that given either objective selection by firm 2, firm 2’s “profit gain” resulting from firm
1’s switching from sales maximization to profit maximization increases with firm 1’s
marginal cost.6 Neutralization of this effect then requires that firm 1 pursue profit
maximization relatively less often as its marginal cost increases. On the other hand, this
profit gain of firm 2 decreases with firm 2’s own marginal cost. Neutralization of this
effect then requires that firm 1 pursue profit maximization relatively more often as firm
2’s marginal cost increases.
2.2. Stage Two Competition is Stackelberg
We now consider the case in which firms engage in a quantity-setting Stackelberg
duopoly game in stage two. For concreteness, we assume that in stage two firm 1, the
When firm 2 pursues profit maximization, its payoff gain from firm 1’a switching from sales
maximization to profit maximization is Π2(Profit,Profit) - Π2(Sales,Profit) = [2C1(A-C2) + 4C12]/9B.
When firm two pursues sales maximization, its corresponding payoff gain is Π 2(Profit,Sales) Π2(Sales,Sales) = C1(2A+C1 - 3C2)/9B.
6
13
leader, decides its output first and then firm 2, the follower, observes firm 1’s output
decision and decides its output. As before, firm 1 and firm 2 select objectives
simultaneously and independently in stage one.
Since firm 1’s output decision is observable to firm 2, firm 1 takes into
consideration firm 2’s reaction to its output when making an output decision in stage two.
Thus we begin our analysis with firm 2’s reaction function in stage two. Let  = (1, 2)
be a pair of objective selections that firm 1 and firm 2 make in stage one. Then, by (1),
given firm 1’s output q1 firm 2’s optimal output is as in (2). By (1) and (2), firm 1’s
optimal output maximizes [A – B(q1+ q 2 (q1)]q1 – λ1C1q1. Assume A  2C1 and A  3C2 .
These conditions guarantee that the resulting Stackelberg equilibrium has positive
quantities for both firms. It follows from simple calculation that quantities and price in
Stackelberg equilibrium are:
q1* 
A  2 C2  21C1 * A  21C1  32 C2
A  21C1  2 C2
, q2 
, P * 
,
2B
4B
4
(4)
Quantities and price in (4) determine firms’ profits associated with objective
selections  = (1, 2). As before, this way we have derived the reduced-form game
between the firms in stage one, in which strategies are objective selections and payoffs
are reduced-form profits associated with such selections.
THEOREM 4: Suppose stage two competition is Stackelberg and suppose A  2C1 and
A  3C2 . Then,
(i) Profit maximization is dominant for firm 1 regardless of the values of C1 and C2.
(ii) Sales maximization is dominant for firm 2 if and only if 6C2  A.
14
Given their objective selections in stage one, as the first-mover in stage two, firm
1 anticipates reaction by the follower to any output choice it may select. In other words,
firm 1 anticipates that firm 2’s output choice is affected by its own output choice
according to firm 2’s reaction curve. This implies that to be optimal firm 1 should
‘correctly’ affect firm 2’s output choice by committing to maximize profit instead of
sales in stage one under all possible cost configurations.
On the other hand, Theorem 4 also implies that being a Stackelberg follower in
stage two does not discourage the pursuit of sales maximization compared to the same
firm’s pursuit of the objective with Cournot competition in stage two. To understand this,
note first when firm 2’s marginal cost satisfies 6C2  A, its reduced-form profit when
either acting as a Cournot firm or acting a Stackelberg follower in stage two is increasing
for small 1 and  2 (see equations (A1) and (A5) in the appendix). This means that with
6C2  A, sales maximization (i.e., 2  0 ) cannot be dominant for firm 2 in either case.
Put differently, 6C2  A must hold for sales maximization to be dominant for firm 2
either acting as a Cournot firm or acting as a Stackelberg follower. However,
when 6C2  A , (A1) and (A5) imply that firm2’s reduced-form profit acting as a
Stackelberg follower equals a positive scalar multiplication of its reduced-form Cournot
profit plus a decreasing function of  2 . Thus, when 6C2  A , sales maximization is a
dominant strategy for both the Stackelberg follower and Cournot firm.
2.3. Stage Two Competition is Bertrand with Differentiated Products
Suppose stage two competition is Bertrand with differentiated products. We
assume demand for firm i’s product is given by
qi  a  bPi  Pj , i  j ,
(5)
15
where Pi is the price set by firm i and Pj is the price set by firm j. We also assume that b >
γ > 0, implying that the effect of a firm’s own price on quantity demanded is greater than
the effect of its rival’s. The above specification implies that goods are substitutes. As
before, firms have linear cost curves TCi = Ciqi.
Given firm i’s objective selection i in stage one, its price decision in stage two is
guided by the following objective function:
OBJi = (Pi - iCi)(a – bPi + Pj).
(6)
The same commitment device as before can be applied to make firm i credibly commit to
pursue objective function as in (6). Assuming interior solution exists, (6) implies that
firm i’s price reaction function in stage two is
Pi  ( Pj )  Pj 
b(i Ci   j C j )
2b  
.
(7)
Now by (7), Bertrand equilibrium price and quantity for firm i are
Pi* 
a  bλi Ci b ( j C j  i Ci )

2b - 
4b 2   2
(8)
and
qi* 
b(a  bλi Ci   j C j )
2b - 

b (b   )( j C j  i Ci )
4b 2   2
.
(9)
Observe that (8) implies that Pi * increases with both λi and λj, while q i*
increases with  j but decreases with λi . Thus, to have interior Nash equilibrium prices
and quantities for both firms regardless of their objective selections, it is necessary and
sufficient that 2(b   )a  (2b 2   2 )Ci for i =1, 2.
16
THEOREM 5: Suppose stage two competition is Bertrand and suppose
2(b   )a  (2b 2   2 )Ci for i =1, 2. Then profit maximization is dominant for each
firm.
This result differs from the outcome with Cournot competition in stage two. With
Cournot competition, quantities are strategic substitutes because a quantity decrease is the
profit-maximizing response to a competitor’s quantity increase. This strategic
substitutability associated with Cournot competition induces firms to be more aggressive
in stage two by committing to pursue an objective different from that of profit
maximization. On the other hand, under Bertrand competition, prices are strategic
complements because a price increase is the profit-maximizing response to a competitor’s
price increase.7 It is this strategic complementarity associated with Bertrand competition
that makes firms less aggressive in stage two by committing to pursue profit
maximization.8
3. A Deterrent Advantage of Sales Maximization
The previous analysis has demonstrated that if firm i’s marginal cost is low relative to
demand and if stage two competition is Cournot, then sales maximization is dominant for
firm i. We now consider an alternative strategic advantage of sales maximization, that of
market deterrence.
Consider the following situation. Firm 1 is the incumbent firm, while firm 2 is the
potential entrant. As entrant, firm 2 pays a fixed cost F to enter the market. We assume
7
The terms strategic complements and strategic substitutes were introduced by Bulow, J., J. Geanakopolos,
and P. Klemperer (1985).
Theorem 5 here is similar to Freshtman and Judd’s Theorem 5. The difference is that in the Freshtman
and Judd model owners always place more than 100% weight on profits in Bertrand equilibrium.
8
17
that when firm 2 enters, the two firms compete in Cournot style. We also assume that
6C1  A, so that sales maximization is dominant for the incumbent.
What we now demonstrate is the decision to pursue sales maximization by firm 1
can also deter entry in the following sense. With firm 1’s marginal cost 6C1  A, there
are values for firm 2’s marginal cost C2 with which firm 2 would not be deterred if firm 1
pursues profit maximization, but would be if firm 1 pursued sales maximization instead.
If so, market deterrence can be profitable to firm 1 for committing to pursue
sales
maximization.
Note first, given firm 1’s objective selection 1  0, (3) implies that firm 2’s profit
with its own objective selection  2 is  *2 (0, 2 )  [ A  (2  3)C2 ][ A  22 C2 ] / 9B.
Thus the optimal objective selection for firm 2 is 2  0 when 6C2  A ;
2  (6C2  A) / 4C2 when 6C2  A .9 Firm 2’s profit with 2  0 is A( A  3C2 ) / 9B and
its profit with 2  (6C2  A) / 4C2 is ( A  2C 2 ) 2 / 8B.
On the other hand, when firm 1 selects 1  1, firm 2’s profit with its own
objective selection  2 is  *2 (1, 2 )  [ A  C1  (2  3)C2 ][ A  C1  22 C2 ] / 9B.
It follows that firm 2’s optimal objective selection is 2  0 when 6 C2  A  C1 ;
2  (6C2  A  C1 ) / 4C2 when 6C2  A  C1 . Firm 2’s profit with the former objective
selection is ( A  C1 )( A  C1  3C2 ) / 9B and its profit with the latter objective selection is
( A  C1  2C2 ) 2 / 8B.
Sales maximization has a deterrent advantage over profit maximization for firm 1
if after entry firm 2’s profits is negative when firm 1 pursues sales maximization, but is
9
We continue to assume 2C2 < A in this section.
18
positive when firm 1 pursues profit maximization. We break the rest of the analysis in
this section in three mutually exclusive but joint exhaustive cases in terms of the values
for firm 2’s marginal cost C2, given firm 1’s marginal cost C1 and parameters A and B in
the demand function.
Case 1: 6C2  A. In this case, sales maximization has the deterrent advantage if and only
if
A( A  3C 2 )
( A  C1 ) ( A  C1  3C 2 )
 F,
 F.
9B
9B
(10)
Combining inequalities in (10) leads to the following condition
A 3BF

 C2 
3
A
A  C1
3BF

.
3
A  C1
(11)
The condition 6C2  A is compatible with (11) if and only if
A 3BF A

 ,
3
A
6
which in turn is equivalent to A  3 2 BF .
Case 2: A  6C2  A  C1. In this case, sales maximization has the deterrent advantage if
and only if
( A  2C 2 ) 2
( A  C1 )( A  C1  3C 2 )
 F,
 F.
8B
9B
(12)
Combining inequalities in (12) yields
A
 2BF  C2 
2
A  C1
3BF

.
3
A  C1
(13)
For the condition A  6C2  A  C1 to be compatible with (13), it suffices to have
19
A  C1 A  C1
A
3BF
 2BF 


,
2
6
3
A  C1
which in turn is equivalent to 3 2 BF  C1  A  3 2 BF  (C1 / 2).
Case 3: A  C1  6C2 . In this case, sales maximization has the deterrent advantage if and
only if
( A  2C 2 ) 2
( A  C1  2C 2 ) 2
 F,
 F.
8B
8B
(14)
Combining the inequalities in (14 ) gives rise to
A
 2BF  C2 
2
A  C1
 2BF .
2
(15)
In this case, the condition A  C1  6C2 is compatible with (15) if and only if
A  C1 A  C1

 2BF ,
6
2
which in turn is equivalent to 3 2 BF  C1  A .
To summarize, the above analysis shows that as long as
3 2 BF  C1  A < 3 2BF , values for firm 2’s marginal cost exist such that sales
maximization has deterrent advantage over profit maximization in the sense as mentioned
above. These values for firm 2’s marginal cost are respectively determined by conditions
6C2  A and (11); A  6C2  A  C1 and (13); and A  C1  6C2 and (15).
4. Concluding Comments
20
The present paper has adopted what we believe is a more direct way of explaining
the prevalence of sales-maximizing behavior of firms. Our approach has been to examine
strategic advantages of maximizing sales in a duopoly setting. The approach involves a
two-stage game. In stage one, firms choose independently and simultaneously as
objectives how much weight to place on profit and how much to weight to place on sales.
In stage two, firms pursue their chosen objectives in a duopoly game. Payoffs were in
terms of profits. The selection of profit versus sales maximization was analyzed under
several common duopoly models.
Under Cournot competition in stage two, sales maximization is dominant for a
firm under the condition that the firm’s marginal cost is low relative to demand. Profit
maximization can never be dominant nor can it be chosen in subgame-perfect
equilibrium. Conditions on demand and costs exist under which firms select mixed profit
and sales objectives in subgame-perfect equilibrium.
Under Stackelberg competition in the second stage, profit maximization is
dominant for the leader. For the follower, however, sales maximization is dominant
under exactly the same conditions as in the case with Cournot competition. Bertrand
competition differed: with differentiated products, profit maximization is always
dominant. As a final exercise we examined the possibility of sales maximization being
used as an entry deterrent by incumbent firms. We concluded sales maximization did
sometimes have deterrent advantage.
Although simple, we believe the analysis has contributed new and valuable
insight into strategic advantages of sales maximization in duopolies. An obvious
extension would be to allow an arbitrary number of firms. Such a model, however, would
21
lead to questions of coalition formation and other sticky problems. We believe that the
basic conclusions of the present paper, that sales maximization can sometimes be
dominant with Cournot competition and as a market deterrent would hold in such a
model.
References
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Behavior by the Type of Control and by Market Power,” Southern Economic
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____________. “Revenue and Profit Maximization: Reply,” Southern Economic
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Baumol, William J., “On the Theory of Oligopoly,” Economica, August 1958, 25, 18798.
Bulow, Jeremy I., John D. Geanakoplos, and Paul D. Klemperer, “Multimarket
Oligopoly: Strategic Substitutes and Complements,” Journal of Political Economy, June
1985, 93, 488-511.
Fershtman, Chaim and Judd, Kenneth L., “Equilibrium Incentives in Oligopoly,”
American Economic Review, December 1987, 77, 927-40.
Heifetz, Aviad, Shannon, Chris, and Speigel, Yossi, “What to Maximize if you Must,”
University of California-Berkeley, mimeo, 2002.
Kreps, David M., A Course in Microeconomic Theory, Princeton, NJ, Princeton
University Press, 1990.
Krouse, Clement G., Theory of Industrial Economics, Basil Blackwell, 1990.
Larson, Alan L., and Giffin, Philip, “Revenue and Profit Maximization: Comment,”
Southern Economic Journal, April 1980, 46, 1221-23.
Maksimovic, V. “Balance Sheet Composition and Value Creation in a Stochastic
Oligopoly, mimeo, 1984.
Peston, M. H., “On the Sales Maximization Hypothesis,” Economica, May 1959, 26,
128-36.
22
Pita, Cristina, “Advanced Notice and Severance Pay Provisions in Labor Contracts,”
Monthly Labor Review, July 1996, 119, 43-50.
Rhode, Paul, and Stegeman, Mark, “Non-Nash Equilibria of Darwinian Dynamics with
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Schelling, Thomas, The Strategy of Conflict. Cambridge, Massachusetts: Harvard
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Vickers, J., “Deregulation and the Theory of the Firm,” Economic Journal Supplement,
1985, 95, 138-47.
Waterson, Michael, Economic Theory of the Industry, Cambridge: Cambridge
University Press, 1984.
23
Proof of Theorems
Theorem 1:
By (3), firm 1’s profit at objective selections  = (1, 2) is given by
 *1 (1, 2 ) 
[ A  (1  3)C1  2 C2 ][ A  21C1  2 C2 ]
9B
(A1)
Thus, fixing  2  [0,1], (A1) implies
[ A  2 C 2  6C1  2(1  '1 )C1 ](1  1 )C1
 ( 1 , 2 )   (1 , 2 ) 
9B
'
*
1
'
*
1
(A2)
for 1 , '1  [0,1]. Thus,
 *1 (0, 2 )   *1 (1 , 2 ), 1  (0,1], 2  [0,1]
 A  2 C 2  6C1  21C1  0, 1  (0,1], 2  [0,1]
 A  6C1  21C1  0, 1  (0,1]
 A  6C1 .
By Symmetry, *2(1, 0) > *2(1, 2 ), 1 [0,1], 2  (0,1]  A  6C2 .
Theorem 2:
By (A2),
 *1 (0,  2 )   *1 (1,  2 ),  2  0,1
 A   2 C 2  4C1 ,  2  0,1
.
 A  4C1 .
By symmetry, *2(1, 0) > *2(1, 1) for 1 = 0, 1 if and only if A > 4C2. This establishes
(i).
By (i), to prove (ii), it only remains to show that
 * i (1,1)   * i (0,0)
24
for i = 1, 2 if and only if A(2Cj – Ci) + (Cj – 2Ci)2 > 0. Note that by (A1), *1(1, 1) = (A +
C2 - 2C1)2/9B and *1(0, 0) = A(A - 3C1)/9B. Thus, *1(1, 1) > *1(0, 0) if and only if
A(2C2 – C1) + (C2 – 2C1)2 > 0. By analogy, *2(1, 1) > *2(0, 0) if and only if A(2C1 –
C2) + (C1 – 2C2)2 > 0.
To prove (iii), note first that by definition firms face a chicken game if and only if
it is optimal for a firm to take a different action from the rival’s. Thus, firms face a
chicken game if and only if *1(1, 0) > *1(0, 0), *1(0, 1) > *1(1, 1), *2(0, 1) >
*2(0, 0), and *2(1, 0) > *2(1, 1). By (A1), *1(1, 0) > *1(0, 0) if and only if 4C1 > A
while *1(0, 1) > *1(1, 1) if and only if 4C1 < A + C2. In summary, *1(1, 0) > *1(0, 0)
and *1(0, 1) > *1(1, 1) if and only if A < 4C1 < A + C2. Similarly, *2(0, 1) > *2(0, 0)
and *2(1, 0) > *2(1, 1) if and only if A < 4C2 < A + C1.
Finally, to prove (iv), note that (A1) implies that
 *1 (1,  2 )   *1 (0,  2 ),  2  0,1
 4C1  A   2 C 2 ,  2  0,1
 4C1  A  C 2 .
By analogy, *2(1, 1) > *2(1, 0) for 1 = 0, 1 if and only if 4C2 > A + C1.
Theorem 3:
Denote by *i the probability that firm i chooses i = 1 in equilibrium. By
Theorem 2(iii), the reduce-form game is a chicken game under the conditions A < 4Ci <
A + Cj for i  j. Hence, 0 < *1 < 1 if and only if 0 < *2 < 1. Given 0 < *2 < 1 , firm 1’s
expected profits with selection 1 is
25
*2*1(1, 1) + (1 – α*2) *1(1, 0). Thus, with 0 < *2 < 1, it must be 0 < *1 < 1 and
 * 2  *1 (0,1)  (1   * 2 ) *1 (0,0)   * 2  *1 (1,1)  (1   * 2 ) *1 (1,0) .
( A3)
By (A1), (A3) implies *2 = (4C1 – A)/C2 . Similarly, *1 = (4C2 – A)/C1.
Theorem 4:
By (4), firms’ profits at objective selections  = (1, 2) are given by
1* (1 , 2 ) 
[ A  (21  4)C1  2 C2 ][ A  2 C2  21C1 ]
, (A4)
8B
and
 *2 
[ A  21C1  (2  4)C2 ][ A  21C1  32 C2 ]
.
16B
(A5)
By (A4),
 1* (1, 2 )   1* (1 , 2 ), 1  [0,1), 2  [0,1]
 4C 21 (1  1 ) 2  0, 1  [0,1)
 4C1  0.
This establishes (i).
By (A5),
 *2 (1 ,0)   *2 (1 ,  2 ), 1  [0,1],  2  (0,1]
 2 A  41C1  12C 2  3 2 C 2 , 1  [0,1],  2  (0,1].
 A  6C 2 .
This establishes (ii).
Theorem 5:
By (8) and (9), for i  j,
Pi
*(1, j )
 Pi
*(i , j )

2b 2 Ci (1  i )
4b 2   2
(A6)
26
and
qi
*(1, j )
 qi
*(i , j )
(1  i )( 2b 2   2 )bCi

.
4b 2   2
(A7)
Thus,
*(i , j )
 *i (1,  j )   *i (i ,  j )  ( pi
 Ci )
(1  i )( 2b 2   2 )bCi 2b 2 Ci (1  i ) *(1, j )

qi
. (A8)
4b 2   2
4b 2   2
Now, by (A6) – (A8),
 1* (1,  2 )   1* (1 ,  2 ), 1  [0,1),  2  [0,1]
 2bq1*(1,2 )  (2b 2   2 )[ P1*(1 ,2 )  C1 ]
 (2b   ) 2 a  (2b 2   2 )C1  b 3  2 C 2  2b 2 (2b 2   2 )1C1  0, 1  [0,1),  2  [0,1]
 (2b   ) 2 a  (2b 2   2 )C1  2b 2 (2b 2   2 )1C1  0, 1  [0,1)
 (2b   ) 2 a  (2b 2   2 )C1 .
By assumption, (2b   ) 2 a  (2b 2   2 )C1 . It thus follows from the above analysis
that 1* (1, 2 )  1* (1 , 2 ), 1  [0,1), 2  [0,1] . Similarly,
 *2 (1 ,1)   *2 (1 , 2 ), 1  [0,1], 2  [0,1) whenever (2b   ) 2 a  (2b 2   2 )C1 .
27